Math Problem Statement
Solution
The problem asks to express the absolute value function as a piecewise-defined function with linear parts.
To break down an absolute value function into a piecewise function, we consider the two cases based on when the expression inside the absolute value is positive or negative.
Step 1: Set up the critical point
The absolute value function changes at the point where the expression inside the absolute value equals zero: This critical point divides the function into two intervals:
- When , the expression inside the absolute value is non-negative, so .
- When , the expression inside the absolute value is negative, so .
Step 2: Write the piecewise function
Thus, the piecewise-defined function is:
\begin{cases} -3x - 4 & \text{if } x < -\frac{4}{3}, \\ 3x + 4 & \text{if } x \geq -\frac{4}{3}. \end{cases}$$ This is the piecewise form of the absolute value function. Would you like more details or have any questions? --- Here are 5 questions to expand your understanding: 1. What does the absolute value represent geometrically on a graph? 2. How do we determine the critical point for an absolute value function? 3. Why does the piecewise function have two different expressions for different values of $$x$$? 4. How would you graph the piecewise function to see the transition at $$x = -\frac{4}{3}$$? 5. Can we generalize this process to any linear absolute value function? **Tip:** When working with absolute value functions, always focus on the point where the expression inside the absolute value equals zero to identify where the function changes behavior.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Piecewise Functions
Absolute Value
Formulas
Absolute value: |a| = a if a >= 0, |a| = -a if a < 0
Piecewise function structure: f(x) = {expression1 if condition1, expression2 if condition2}
Theorems
Definition of absolute value
Suitable Grade Level
Grades 9-12